Tautological Entailments - CiteSeerX

Tautological Entailments Author(s): Alan Ross Anderson and Nuel D. Belnap, Jr. Source: Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition, Vol. 13, No. 1/2 (Jan. - Feb., 1962), pp. 9-24 Published by: Springer Stable URL: http://www.jstor.org/stable/4318404 Accessed: 28/05/2009 15:09 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=springer. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected].

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Entailments Tautological by ALAN ROSS ANDERSON and NUEL D. BELNAP, JR. YALE UNIVERSITY

IN The PureCalculusof Entailment' we offereda formaltheoryconcerning

the relationbetweenA and B when B followslogicallyfrom A. We showed there that the calculusEI avoidedcertainfallaciesof modalityand fallacies of relevancewhich are allowed as valid inferencesby those who hold that material,intuitionistic,and strict "implication"relationsare implicationrelations.We takeit as our problemin this paperto extendthe previousresults by findingplausiblecriteriafor pickingout from among first-degreeentailments (i.e., entailmentsof the form A->B, whereA and B are purelytruthfunctional) those that arevalid.We referto such validentailmentsas tautologicalentailments. Clearlynone of the "implication"relationsmentionedabovewill do as a criterion,since if any of these conditionswere sufficientfor entailment,we would have A&-.-A-->B.But as the informaldiscussionof the paperscited in note 1 would indicate, we regarda contradictionA&- A as in general irrelevantto an arbitrarypropositionB, and we accordinglythink of the principle"(A and not-A) implies B" as embodyinga fallacyof relevance. On the otherhand,"(A and not-A) impliesA" seemstrueto our preanalytic idea of conjunction,since it is a specialcase of the plausibleprinciple"(A and B) implies A." What is wanted, inter alia, is a way of distinguishing these cases. Von Wright (27) has proposeda criterionof the sortwe seek: "A entails B, if and only if, by meansof logic, it is possibleto come to know the truth of A D B without coming to know the falsehoodof A or the truth of B." Geach (14), following von Wright, proposesa slightly differentcriterion: "I maintainthat A entailsB if and only if there is an a prioriway of getting to know that A D B which is not a way of getting to know whether A or whetherB." These proposalsseem to us to be on the right track,but they need improvement,fortwo reasons. In the first place, the expression"come to know" is vague. One might imaginea person's"comingto know"the truthof A D Bv-B without coming to know the truth of Bv-B owing to the fact (say) that the formula wasfed into a computerprogrammedto test tautologies.Strawson(26) finds a similardifficulty: "It appearsthat von Wright has overlookedthe implicationsof one faAUTHORS' NOTE.

for thispaperwassupportedin partby the Officeof Naval The research

Research,Contractno. SAR/Nonr-609(16), GroupPsychologyBranch.

9

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miliarwayof arrivingat the paradoxes.Considerp D (qv#-.wq).Von Wright would of coursewish to deny that p entailsqv-q. Now the followingis demonstrableindependently of demonstratingthe falsity of qv--q or, of course,the truthof p: (1) p D ((p&q)v(p&.-q));

for in the truth-tableproofof (1) there is only one 'unmixed'column (i.e. columnconsistingpurelyof Ts or Fs) which is the last column showingthe whole expressionto be a tautology.Still more obviouslythe followingare demonstrableindependentlyof demonstratingthe truth of p or the falsity of qvq: (2) ( (p&q)v(p&~q) ) D (p&(qv~q) ) (3) (p&(qv~q)) D (qv~q) (4) (p :Dq) D [(q D r) D ((r D s) D (p D s))]. For although (2) and (3) both contain qv~q, they are respectivelysubstitution instances of ((p&q)v(p&r)) D (p&(qvr)) and of (p&q) D q. Hence, by substitutingthe second halvesof (1), (2), and (3) for q, r, and s respectivelyin (4) and by repeatedapplicationsof Modus Ponens, we obtain a demonstrationof p D (qv~q) which is independentof demonstratingthe falsityof p or the truthof qv~q. Consequently,on von Wright's definition,p entailsqv~q. But this is one of the paradoxicalcaseswhich his theoryis intendedto avoid." In reply,von Wright (28) distinguishesseveralsensesof "comingto know," in one of which, he claims, Strawson'salleged counterexampleis not a counterexample.But we do not pursuethese distinctions,because,as von Wright indicates,the situation remainsin an unsatisfactorilyvague state. Smiley (25), while retainingthe spiritof von Wright'sproposal,modifies it in such a way as to eliminate the vagueness. He holds that A1&. . . &A, should entail B just in case (A1& . . . &A,n)D B is a substitution instance of a tautology (A1'&. . . &An')D B', such that neither B' nor the denial of A,'& . . . &An'is provable. He cites as an example: " . . . for any A, A&

~A entailsA, becauseA&~A-->Ais a substitutioninstanceof A&B->A;but A&-A does not entail just any B, becausethere is in generalno way of derivingA&~A-AB froman implicationwhich is itself tautologousbut whose antecedentis not self-contradictory." Smiley'scriteriongives rise to a definitionof entailmentwhich is effectively decidable,and seems also to capturethe intent of von Wright and Geach. But there is an applicationof it which leads to a second objection (as do the proposalsof von Wright and Geach, underat least one interpretation). Since A->A&(Bv~B) satisfiesthe criterion,and A&(Bv,-B)->

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Bv,...B does also, we find that the paradoxA->Bv~B can be avoidedonly at the priceof giving up transitivityof entailment.This unwelcomecourse has in factbeen recommendedby Lewy (19), Geach (14), and Smiley (25). Smileyconsidersthe matteras follows: "It is true that 'connexion of meanings'is not as simple as might be thought: 'it has been plausiblyarguedthat any propositionasserts(at least implicitly) something about all objects whatsoever."Grassis green," for instance,saysamongotherthingsthat it is not the casethat grassis not-green androsesarered,and so on. This followssimplyfromthe factthat any proposition constitutesa denial of some other propositionsand thereforeof all conjunctionsof which these propositionsaremembers.'[Bennett (11).] But to concludefrom this that 'thus there is a connexionof meaningsbetween any two propositions;and a necessaryor impossiblepropositionhas with any other propositiona connexionof meaningssuch as will validateone or other of the paradoxicalinferences'is to assumethat 'connexionof meanings'is a transitiverelation,and it is only necessaryto examinethe derivationof one of the paradoxicalprinciplesto see that it is not." It is of coursecorrectthat 'connexionof meanings'is not transitive,at least under one interpretation:there is a meaning connection between A and A&B,and also betweenA&Band B - but there need be no connection of meaningbetween A and B. And what this shows is that connection of meaning,thoughnecessary,is not a sufficientconditionfor entailment,since the latterrelationis transitive.Any criterionaccordingto which entailment is not transitive,is ipso facto wrong.It seems in fact incrediblethat anyone should admit that A entails B, and that B entails C, but feel that some furtherargumentwas requiredto establishthat A entails C. What better evidenceforA->C couldone want? The failureof these proposalsarisesfroman attempt to applythem to all formulas.For there is a class of entailmentsfor which Smiley'scriterionis absolutelyunarguablyboth a necessaryand a sufficientcondition;namely, the class of primitiveentailments,which, after introducingsome auxiliary notions,we proceedto define. An atom is a propositionalvariableor the negate of a propositionalvariable. A primitivedisjunctionis a disjunctionA1vA2. . . vAm,where each disjunctA, is an atom. A primitiveconjunctionis a conjunctionB1&B2. . . &Bn,eachBjbeingan atom.A->B is a primitiveentailmentif A is a primitive conjunctionand B is a primitivedisjunction.We take it as obviousthat if A and B areboth atoms,then A->B shouldbe a validentailmentif and only if A and B are the sameatom; e.g., we would want p-?p and p-- p but not p-*q or p-~-~-p.We think it equallyobvious that if A1&A2. . . &A is a primitive conjunction and B1v . . . vBn is a primitive disjunction, then

12 A,& . .. &Am-Biv

STUDIES PHILOSOPHICAL .

..

vB,, should be a valid entailment if and only if

some atom Ai is the same as some atom Bj; e.g., we want p&q-*qvr,and -p&q&r-sv,pv,r, but neither p&-q-->r, nor -p-*qvpvr, nor (it need hardlybe added) p&,pq. We shallsay that a primitiveentailmentA- >B is explicitlytautological,if some (conjoined) atom of A is identical with some (disjoined) atom of B. Such entailmentsmay be thought of as satisfying the classicaldogmathat for A to entail B, B must be "contained"in A.2 It is clearthat explicitlytautologicalentailmentssatisfythe requirements of von Wright, Geach, and Smiley: everyexplicitlytautologicalentailment answersto a material"implication"which is a substitutioninstance of a tautologousmaterial"implication"with noncontradictoryantecedent and nontautologousconsequent;and evidentlywe mayascertainthe truth of the entailmentwithout coming to know the truth of the consequentor the falsity of the antecedent.Certainlyall explicitlytautologicalentailmentsare valid,and we see absolutelyno way in which the stockof valid primitiveentailments could plausiblybe enlarged;we take it thereforethat explicitly tautologicalentailmenthoodis both necessaryand sufficientfor the validity of a primitiveentailment. Let us now returnto a considerationof the (nonprimitive) entailment A-->A&(Bv,B), which, as was pointed out before, satisfiesthe criteriaof von Wright, Geach, and Smiley. Lewy (19) remarks(in effect) that A-A&(Bv,- B) seems "verynearly,if not quite," as counterintuitiveas A(Bv~B). We agreein substancewith Lewy,but we think his estimateis too high: A-*A&(Bv~ B) is exactly50 percent as counterintuitiveas A >(Bv~ B). That is, A--A&(Bv~B) is valid just in case both the primitiveentailments A-->Aand A--*(Bv- B) arevalid;the formeris valid,but the latteris not-hence A--A&(Bv~B) does not representa valid inference.Dually, (A&~A)vB->B is valid if and only if both A&A-->B and B--B are valid; andagainone is validand the othernot. These considerationssuggestcriteriafor evaluatingcertainfirst-degreeentailmentsother than primitiveones: A-*B&C is valid if and only if both A->B and A--C are valid;and AvB-*C is valid if and only if A-*C and B-->Careboth valid.This givesus a techniquefor evaluatingentailmentsin normal form, i.e., entailmentsA--B having the form A,v . . . vA-->Bl & . . . &Bm,where each Ai is a primitive conjunction and each Bj is a primitive disjunction.Such an entailmentis valid just in case each Ai->B is explicitly tautological.For example, (p&q)v p-> (~pvp) &( ~ pvq), (p&q)v (p&r)--p&(qvr),(pvq)vr->pv(qvr),and p&q->q&(rvp), areall validentailments in normalform;but the following are invalid: (p&~p)vq-*q, and p-*p&(qv#~q)The proposalas statedis still not complete,however,sincetherearecombi-

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nationsof disjunctionand conjunctionwhich the rule fails to cover;there is no way to applyit directlyto A&Q( ..AvB)->B or (A&B), for example. But all that would be requiredto make the criterioneverywhere applicableis the ability to convertany first-degreeentailmentinto normal form. This in turn will requireconvertingtruth-functionalformulasinto disjunctiveand conjunctivenormal forms (i.e., to disjunctionsof one or more primitiveconjunctions,and to conjunctionsof one or more primitive disjunctions). We therefore propose adding the following replacementrules (all of which we take to preservevalidity), which enable us to find, for any firstdegreeentailment,an equivalententailmentin normalform: Commutativity:replacea partA&Bby B&A;replacea partAvB by BvA; Associativity:replacea part (A&B)&Cby A&(B&C), and conversely;replacea part (AvB)vC by Av(BvC), and conversely; Distributivity:replace a part A&(BvC) by (A&B)v(A&C), and conversely;replacea partAv(B&C) by (AvB)&(AvC) and conversely; Double negation:replacea wf partA by o-A and conversely; De Morgan'slaws:replacea part (A&B) by -AvB, and conversely; replacea part - (AvB) by -A&-B, and conversely. We proposethen to call an entailmentA--B, whereA and B are purely truth-functional,a tautologicalentailment,if A->B has a normalform3Alv . . .vA,-->Bi& . . . &Bmsuch that each A,->Bjis an explicitlytautological entailment. We proposetautologicalentailmenthoodas a necessaryand sufficientcondition for the validityof first-degree entailments.(The propertyis obviously decidable.) As an example,we show that (p D q)&(q D r)->(p D r) is invalid.By the definitionof "D ," we have (- pvq)&(~-qvr)->~ pvrwhich has a normal form, (~p&~q)v(,-p&r)v(q&q)v(q&r)pvr. But q&~q--.~pvr is not an explicitlytautologicalentailment;hence the candidatefails. The foregoingexampleshowsthat material"implication"is not transitive, if by sayingthat R is transitivewe mean that ARB and BRC jointly entail ARC. The differencesbetweentautologicalentailmentsand other allegedimplication relationsbetween truth-functionscan be broughtout clearlyby consideringthe primitiveentailments to which an arbitrarycandidateA->B reduces.If we regardA-*B as valid only when each Ar-*Bjis an explicitly tautologicalentailment,then A->B is a validentailment.If we alsocallA-*B "valid"when each AX->Bjeither (i) is an explicitlytautologicalentailment or (ii) contains atoms C and ~C conjoinedin the antecedent,then the

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first-degreefragmentof the propositionalcalculusof Fitch (13) is complete and sound for this definitionof "validity."And finallyif we add to (i) and (ii) the condition (iii) that A,--Bj is "valid"if Bj containsC and ~C disjointly,then the arrowreducesto material"implication."(So faras we know, no one has investigatedthe systemobtainedby takingas "valid"primitive entailmentssatisfying(i) and (iii) only.) TautologicalEntailmentsand E The systemE of entailment(3) maybe capturedaxiomaticallyas follows: Axiomschematafor the systemE of entailment: Entailment. E.1 A-*A->B->B E.2 A--B-.B--C->.A-4C E.3 (A->.A->B)->.A->B Conjunction. E.4 A&B->A E.5 A&B-4B E.6 (A-B ) &(A-C )->.A--(B&C ) Relatingmodalityand conjunction. E.7 NA&NB->N(A&B) [NA = dfA>A-A.] Disjunction. E.8 A--AvB E.9 B->AvB E.10 (A->C) &(B--C)-->(AvB)--C Relatingconjunctionand disjunction. E.1l A&(BvC)-(A&B)vC Negation. E.12 A-A--A E.13 A-->B-.B -->A ~A-A E.14 Rules: Modus ponens: If A-->Bis asserted,then from A to infer B. Adjunction:FromA and B to inferA&B. It developsthat the system E is sound and complete relativelyto tautological entailmentsin the followingsense: Theorem.A first-degreeentailmentA--.B (that is, where A and B contain only variables, ', v, and &) is provablein E if and only if A->B is a tautologicalentailment(Belnap,10).

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Proof. By methodsof Ackermann(1), it can easilybe shown that a firstdegreeentailmentA-+B is provablein E just in casea normalformA1v . vAm*Bl& . . . &Bnof A-4B is provablein E. (All replacementrules used in definingtautologicalentailmentsare provableas co-entailmentsin E, and E has a derivablereplacementrule.) Then by E.2, E.4-E.6, and E.8-E.10, A->B is provablein E just in case each A,--Bj is provablein E; and A--*B is a tautologicalentailmentjust in case each AX->Bjis an explicitlytautological entailment.Hence it will sufficeto show that each primitiveentailment A--*Bj is provableif and only if it is an explicitlytautologicalentailment. It is trivialthat all explicitlytautologicalprimitiveentailmentsare provablein E, and we considerthe converse.For this we need the accompanyingmatrices,whichsatisfythe axiomsand rulesof E. -3 -2 -1 -3 +3 +3-+33 -2-3 +2 -3 -1-3 -3 +1 -0 -3 -3 -3 +0 -3 -2 -1 +1-3-3-1 +2 -3 -2 -3 +3 -3-3-3

-3 -2 -1 -O +0 +1 +2 +3

-3 -3 -3 -3 -3 -3 -3 -3 -3

-2 -3 -2 -3 -2 -3 -3 -2 -2

-1 -3 -3 -1 -1 -3 -1 -3 -1

A&B -O +0 -3 -3 -2 -3 -1 -3 -O -3 -3 +0 -1 +0 -2 +0 -0 +0

A--B -0 +0 +3 +2-3-3 +1 -3 +0 -3 -0 +0 -1 -3 -2 -3 -3 -3

~A +1 +2 +3 +3 ?2 +1 -3 -3 -3 +1 +2 +1 -3 -3 +2 -3 -3

+3 +3 +3 +3 +3 +3 +3 +3 +3

-3 -3 -3 -2 -2 -1 -1 -0 -0 +0 +0 +1 +1 +2 +2 +3+3+

-2 -2 -2 -0 -0 +2 +3 +2

-3 +3 -2 +2 -1 +1 -0 +0 +0 -0 +1-1 +2-2 +3 -3 AvB

+1 -3 -3 -1 -1 +0 +1 +0 +1

+2 -3 -2 -3 -2 +0 +0 +2 +2

+3 -3 -2 -1 -O +0 +1 +2 +3

-1 -1 -0 -1 -0 +1 +1 +3 3

-O +0 +1 +2 +3 -0 -0 -0 -O +3 +3 +3 +3

+0 +2 +1 +3 +0 +1 +2 3+3

+1 +3 +1 +3 +1 +1 +3

+2 +2 +3 +3 +2 +3 +2 +3

+3 +3 +3 +3 +3 +3 +3 +3

Let A-->Bjbe a primitiveentailmentwhich is not explicitlytautological. Then assignvaluesfromthe accompanyingmatrixto the variablesin A-->B, as follows: (i) If the propositionalvariablep occursin Ai but not in Bj, give p the value+1; (ii) If p occursin Bjbut not in Ai, givep the value+2; (iii) If p occursin Ai, and -p occursin Bj, give p the value +3; and (iv) If -p occursin A, and p occursin Bj, give p the value -3. (We note that (iii) and (iv) cannot conflict, since A--*Bj is not an explicitlytautologicalentailment,and hence Ai and Bj shareno atoms.)

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Underthis assignment,Ai alwaysassumesa value? 1 or + 3, and B, always assumesa value? 2 or -3. HenceA,--B, assumesthe value- 3, and is therefore unprovable(plus valuesbeing designated). The matricesalso enable us to show that E satisfiesa plausiblenecessary condition for avoidanceof fallaciesof relevance: Theorem.A--B is provablein E only if A and B sharea variable(Belnap, 8, 9). If A and B fail to sharea variable,then we may assignall variablesof A the value +?1, so that A takesthe value -+-1, and we may assignall variables of B the value+2, so that B takesthe value ?2. Then A->B takesthe value 3, and is thereforeunprovable. The matricesabove may be given the following partialinterpretation.4 Choose A and B in such a way that A->B, ~B->B, ~AB->~B, ,- BA-+ ~A, and ~ (A~A->Bv~B). (These conditionsmay be securedby letting B be CvCD, and A be ~DvCD, where C is pv~p and D is qv~q; and wherep is "Napoleonwas born in Corsica,"and q is "666 is a perfectnumber"and hence ~ (p-p---qv~-q).) Then we assignpropositionalvaluesto v(i) to the numbersi as follows: v(-3) =-A&~B v(+O) _ A&B ~A v(-2) v(+1)-B A v(-1) = B v(+2) AvB B v(-O) v(+3) AvThis gives a complete interpretationof the tables for conjunction,disjunction, and negation,and a partialinterpretationof the table for entailment, as follows. For conjunction,i&j - k just in case v(i)&v(j)--v(k) and v(k) -*v(i)&v(j) aretautologicalentailments.Similarlyfor disjunctionand negation. For entailments,we read plus values in the matrixas "true,"minus valuesas "false." We note in passingthat necessaryand sufficientconditionsfor ~A--A amongfirst-degreeentailmentsare as follows: Theorem.Let B be a truth-function.Then ~B-->Bis valid if and only if a conjunctive normal form B1&. . . &B. of B has the property that for each Bi and Bj, thereis an atom p such that p occursin Bi I Bj and -p occursin Bj Bi . The proofis left to the reader. .

Truth-Functional Fallaciesof Relevance As logicianshave alwaystaught,logic is a formalmatter,and the validity of an inferencehas nothing to do with the truth or falsityof premise(s)or

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conclusion.But the view that material"implication"is an implicationrelation fliessquarelyin the faceof this teaching,and leadsdirectlyand immediately to fallaciesof relevance.A&~A---B has been defendedon the ground that althoughit is useless,it is harmless,since the antecedentcan neverbe realized.We grantthat it is harmlessin this sense, but still contend that it is harmfulin anothersense, namely,in being false. To be sure, there is a (somewhatodd) sensein which we "losecontrol"in the presenceof contradictions.Namely: we definea manifestrepugnancyas a primitiveconjunction A1&. . . &A, havingthe propertythat for each Ai, if A, is a propositional variable,then for some Aj, A> A, and if A, is the negate of a propositionalvariable,then for some A;, Ai -= --Aj. An exampleis p& -p& q&,q&r& - r. And for such expressionswe have the following: Theorem. Manifest repugnanciesentail every truth function to which they areanalyticallyrelevant. Proof.We saythat A is analyticallyrelevantto B if all variablesof B occur in A (see note 2). The theoremthen states that a manifestrepugnancyentails everytruth-functionalcompoundof its own variables.And this may be readilyseen as follows. Let A be a manifestrepugnancy,and let B be any truth-functionof the variablesin A. RewriteB equivalentlyin conjunctive normalformB1&. . . &B..Then each Bi containsat least one of the atoms in A; hence each A--i is an explicitlytautologicalentailment. Dually, we have that every truth-functionalexpressionentails a (very weak) tautology, consisting of a disjunction of various special cases of Av'--A.But admittingtheseobviouslogicaltruthsis a farcryfromadmitting that a contradictionentailsanyold thing. It is of course sometimessaid that the implicationwe use admits that falseor contradictorypropositionsimplyanythingyou like, and we aregiven the example"If Trumanwins, I'll be a monkey'suncle."But it seems to us unsatisfactoryto dignify as a principleof logic what is obviouslyno more than a rhetoricalfigureof speech,and a facetiousone at that; one might as well cite Cicero'suse of praeteritioas evidencethat one can do and not do the same thing at the same time (and in the same respect). Lewis,however,has explicitlyarguedthat the paradoxesof strict "implication" "state a fact about deducibility"(18, p. 251) and has presented "independentproofs"of their validity.Since there is a clearoppositionbetween our position and that of Lewis (and practicallyeveryoneelse), we will examineone of these proofsin detail. The argumentconcernsAA--->B, and has two steps, (i) "A entails B" or "A--B" meansthat B is deduciblefrom A "by some mode of inference which is valid" (18, p. 248), and (ii) there is a "validmode of inference" fromA~A to B.

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We may accept (i) without cavil. Argumentsfor (ii), that is, for the propositionthat there is a valid mode of inferencefrom a contradictionto any arbitraryproposition,were known to severallogiciansflourishingcirca the year1350,and are found in extantwritingsof the astute Bishop of Halberstadt,Albertof Saxony(see Boehner(12), pp. 99-100). Lewisand Langford'spresentationof the argument(18, p. 250) does not differsignificantly from Albert's,althoughit is almost certainthat the modernappearanceof the argumentrepresentsa rediscoveryratherthan a continuityof tradition. The argumenthasalsobeen acceptedby a varietyof othermodernlogicianse.g., Popper (21), pp. 407-10, and (22) -and indeed,as Bennett points out (11, p. 451), "this acceptancehas not been an entirely academicmatter. William Kneale [17] and Popper[23] have both used the paradoxesas integralpartsof theirrespectiveaccountsof the natureof logic." Though departingin insignificantdetailfromLewis'own, the followingis a convenient presentationof the argument.Grant that the following are 'validmodesof inference': 1. fromA&Bto inferA, 2. fromA&Bto inferB, 3. fromA to inferAvB,and 4. fromAvBand ,-A to inferB. The argumentthen proceedsin this way: (a) A&,..A premise (b) A from (a) by I (c) AvB from (b) by 3 (d) -..A from (a) by 2 (e) B conclusion: from (c) and (d) by 4. Than which nothing could be simpler: if the four rules above are "valid modes of inference"and if "A-4B" meansthat there is a valid mode of inferencefromA to B, then a contradictionsuch as A&,..,Asurelydoes entail any arbitraryproposition,B, whence A&-..A->Brepresentsa fact about deducibility. We agreewith those who findthe argumentfrom (a) to (e) self-evidently preposterous,and fromthe point of view we advocateit is immediatelyobvious wherethe fallaciousstep occurs:namely,in passingfrom (c) and (d) to (e). The principle4 (fromAvB and -A to inferB), which commitsa fallacy of relevance,is not a tautologicalentailment.We thereforereject 4 as an entailment,andas a validprincipleof inference. We seem to have been pushed into one of the "peculiarpositions"of which Priorspeaks(24, p. 195), forwe areexplicitlydenyingthat the principle of the disjunctivesyllogism-or detachmentfor material"implication"-

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is a "validmode of inference."The validityof this formof inferenceis something Lewis neverdoubts (see, for example,Lewisand Langford(18), pp. 242-43) and is something which has perhapsnever been seriouslyquestioned before (though the possibilityof dispensingwith the disjunctivesyllogismis raisedby Smiley (25) ). Nevertheless,we do hold that the inference from ,,A and AvB to B is an error: it is a simple inferentialmistake.Such an inferencecommitsnothingless than a fallacyof relevance.We shall first of this thesis and then proceedto an anticipatepossiblemisinterpretations "independentproof"of the invalidityof A (AvB)--B. In the first place, we do not deny that the inferencefrom I-,-A and 1AvB to I-B is valid,where "I-A"means 'A is a theoremof the two-valued propositionalcalculus.'However,fromthis it does not follow that B follows from -A and AvB, nor does it follow that if I-,A and I-AvB then B follows from ~AandAvB.We evenadmitthat if I-B then B is necessarilytrue, and still hold that the argumentfrom ~Aand AvB to B is invalidevenwhen I-,--,Aand I-AvB (and hence I-B). Such a claim would be senselesson Lewis'doctrine,for to admit B is necessarilytrue is to admit that any argument for B is valid. Secondly,we do not say that the inferencefrom .A and AvB to B is invalidfor all choicesof A and B; it will be validat least when A entailsB (in which case -.A is not required)or when --A entailsB (in which case AvB is not required);more generally,it will be valid when A- A entails B (in whichcasethe disjoinedpremiseB is not required). Furthermore,in rejectingthe principleof the disjunctivesyllogism,we intend to restrictour rejectionto the case in which the "or"is taken truthfunctionally.In generaland with respect to our ordinaryreasoningsthis wouldnot be the case;perhapsalwayswhen the principleis usedin reasoning one has in mind an intensionalmeaningof "or,"where there is relevance between the disjuncts.But for the intensionalmeaning of "or,"it seems clearthat the analoguesof A-*AvB are invalid,since this would hold only if B wasrelevantto A; hence, thereis a sense in which the realflawin Lewis' argumentis not a fallacyof relevancebut rathera fallacyof ambiguity: the passagefrom (b) to (c) is valid only if the "v" is read truth-functionally, while the passagefrom (c) and (d) to (e) is valid only if the "v" is taken intensionally.We shall furtherconsiderthe intensional"or"below. Our finalremarkconcernswhat Lewis might have meant by "somevalid formof inference."It is hardlylikelythat he meant that a form of inference is valid if and only if either the premisesare false or the conclusiontrue ("materialvalidity");more plausibly,he might have meant that a form of inferenceis valid if and only if it is necessarythat either the premisesare false or the conclusiontrue ("strictvalidity").If this is what Lewis meant,

20

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then we agreeat once that the inferencefromA and -..AvBto B is validin this sense. However,if this is all that Lewis meant by "somevalid form of inference,"then his long argumentfor A&.A->B is a quite unnecessary detour,for in this sense we should have agreedat once that there is a valid form of inferencefrom A&,..,Ato B: it is surelytrue that necessarilyeither the premiseis false or the conclusion is true inasmuchas the premiseis necessarilyfalse. In short,Lewis'"independentproof"of A& A--B is convincing if "validinference"is definedin terms of strict implication;but in that caseit is superfluousand circular.And his argumentservesa usefulpurpose only if "validinference"is thought of in some other sense, in which casehe hasfailedto prove-or evento arguefor-his premises.Finally,should he wish to escapethe horns of this dilemmaby remarkingthat the various forms of inferenceused in the argumentare valid in the sense of having alwaysbeen acceptedand used without question, then we should rest our case on the fallacyof ambiguitynoted above. Such a thesis so stronglystated will seem hopelesslynaive to those logicians whose logical intuitions have been numbed throughhearingand repeatingthe logicians'fairytales of the past half century,and hence stands in need of furthersupport.It will be insistedthat to deny detachmentfor materialand strict implication,as well as to deny the principleof the disjunctivesyllogism,surelygoes too far: 'from A and AvB to infer B,' for example,is surelyvalid. For one of the premisesstates that at least one of A and B is true, and since the other premise, -A, saysthat A can't be the true one, the true one must be B (see Popper (22), p. 48). Our reply is to remarkagainthat this argumentcommitsa fallacyof ambiguity.There are indeedimportantsensesof "or,""atleast one,"etc., for which the argument from -A and A-or-Bis perfectlyvalid,namely,sensesin which thereis a true relevancebetweenA and B, for example,the sense in which "A-or-B"means preciselythat -A entails B. However,in this sense of "or,"the inference from A to A-or-Bis fallacious,and thereforethis sense of "or"is not preserved in the truth-functionalconstant translatedby the same word. As Lewishimself arguedin some earlyarticles,there are intensionalmeanings of "or," "not both," "at least one is true," etc., as well as of "if

. . .

then

Those who claim that only an intensionalsense of these words will supportinferencesareright-Lewis' only errorwas in supposinghe captured this senseby tackinga modaloperatoronto a fundamentallytruth-functional formula. Nevertheless,the inferencefrom -.,Aand A-or-Bto B is sometimesvalid even when the "or"is truth-functional,for it will be valid in everycase in which A,-A-->B.For example,althoughthe decisionprocedureof the previis not valid, neverous section shows that S(AB)ABvABvAB ."

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to infer theless the inferencefrom (AB) and ABv,ABvA,Bv-A-B ,ABvA-Bv.A,B is perfectlyvalidsince,as is easilyverified,e (AB)AB >ABvA,Bv-A,.'B. In general,if A1vA2v. . . vAn (n 2k) is a Boolean expansion,then the inferencefrom A1vA2v. . . vA,,and -- A1 to A2v ... vAnis valid,as the followingconsiderationsshow. Lemma. Let A, and B1 .

.

. B,,, all be primitive conjunctions (i.e., con-

junctionsof atoms, where an atom is either a propositionalvariableor the .negate of a propositional variable). Then A-- (B1v . . . vB,,m)is a theorem

if and only if for some Bi, A->Bi is a theorem. Proof.Trivially,if for some Bi, A->Bi is provable,then so is A-(Blv vBm). For the converse,supposethat for no Bi is A->Bi provable.Then every Bi containssome atom not containedin A. Hence the conjunctivenormal form C1& . . . &Cnof B1v . . . vBmcontains a disjunction Cj, none of the

atoms of which occur in A; so A-*Cj is not provable.But A- (B1v . . . vBm)is provable if and only if A-- (C1& . . . &C,,) is provable, i.e., if and only if A--Cj is provable for each j. Hence A-* (B1v . . . vBm) is unprov-

able, as required. Theorem. Let pi . . . pn be distinct propositional letters, and let A, B1 . . . BM,be conjunctions of the form pl'&p2' . . . &p,/ where each pi' is either pt or pi. Then A&-A-*(B1v . . . vBm) is provable if and only if for each i (1 -1 i n) there is a Bj among B1 . . . Bm which differs from

A in at most its i-th conjunct. Proof.ConsiderA&,A, which has the formpl'&p2'& .

. . &p,t&(p1"Vp2'f

v . . . vpmt"),where pi" is - pi if pi' is pi, and pi" is pi if pi' is , pi. It is seen that this expression entails Blv . . .vBm if and only if we have, for each -- (B1v . . .vBm). By the lemma, this entailment i, (pi'& . . . &pn'&pi") holds if and only if (pi'& . . . &pn'&pi") --Bj, for some j. But this can hold if and only if some Bj is pi'& . . . &pi'&. . . pn' or pi'& . . . &pi"& . . . pn'

and hence differsfromA in at most the i-th conjunct. Since everytautologicalBooleandisjunctivenormalformsatisfiesthe condition of the theorem, it follows then that if A1v . . . vAn is a tautology in Boolean disjunctive normal form, then -,A1 (Alv . . . vAn)-*A2v . . . vAn

is provable.But of coursein the generalcase, A (AvB)->B is rejected,since (herecomesthe "independentproof"promisedabove) A(AvB)->B if and only if ~AAv,AB--B, only if -AA->B, which is absurd. Notice that negation,which is at the bottom of all truth-functionalfallacies of relevance,playsa veryweakrole in the theoryof entailmentsbetween truth-functions: if A1v . . . vA,->Bi& . . . &Bmis such an entailment in normalform,then all negationsignsoccurringin the formulamaybe deleted without affectingvalidity.This featureof the situationreinforcesour claim

22

PHILOSOPHICAL STUDIES (which standsin fact at the core of the traditionin formallogic), that the validityof a valid entailmentdependsin no way on the truth or falsity of antecedentor consequent. And as finalevidencefor our contentionwe make the followingobservations: The truth of A-or-B,with truth-functional"or,"is not a sufficientcondition for the truthof "If it werenot the casethat A, then it wouldbe the case that B." Example:It is truethat eitherNapoleonwasborn in Corsicaor else the numberof the beast is perfect (with truth-functional"or");but it does not followthat had Napoleonnot been bornin Corsica,666 wouldequalthe sum of its factors.On the otherhand the intensionalvarietiesof "or"which do supportthe disjunctivesyllogismare such as to supportcorresponding (possiblycounterfactual)subjunctiveconditionals.When one says"that is either Drosophiliamelanogasteror D. virilis,I'm not sure which," and on finding that it wasn'tD. melanogaster,concludesthat it was D. virilis,no fallacyis being committed.But this is preciselybecause"or"in this context means "if it isn't one, then it is the other."Of coursethere is no question here of a relationof logical entailment (which has been our principalinterest); evidently some other sense of "if . . . then . . ." is involved. But

it shouldbe equallyclearthat it is not simplythe truth-functional"or"either, fromthe fact thata speakerwouldnaturallyfeel that if whathe saidwastrue, then if it hadn'tbeen D. virilis,it would havebeen D. melanogaster.And in the sense of "or"involved, it does not follow from the fact that it is D. virilisthat it is eitherD. melanogasteror D. virilis-no more than it follows solelyfromthe fact that it wasD. virilis,that if it hadn'tbeen, it wouldhave been D. melanogaster. The logical differenceswe have been discussingare subtle, and we think it is difficultor impossibleto give conclusiveevidence favoringthe distinctions among the varioussenses of "or" we have been considering.But whetheror not the readeris in sympathywith our views,it might still be of interestto find a case (if such exists) wherea personother than a logician makingjokesseriouslyholds a propositionA-or-B,in a sense warrantinginferenceof B with the additionalpremisenot-A,but is unwillingto admitany subjunctiveconclusionfromA-or-B.If no such examplesexist, then we will feel we havemadeour case (and if examplesdo exist,we reservethe rightto tryto findsomethingfunnyaboutthem). ReceivedSeptember12, 1960 NOTES 1Andersonand Belnap (5). The axiomsfor the pure calculusEI of entailmentare A A-)B-->B, A--B--->.B-->C-.A-->C, and (A->.A--->B)-->.A->B, with modus ponens as

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sole rule.Furtherinformation maybe foundin Anderson(2), Andersonand Belnap(3, 4), Anderson,Belnap,andWallace(6), and Belnap(7, 8, 9, 10). In the presentpaperwe use -> forentailment,v fordisjunction, and &forconjunction, fornegation.Whereconvenientwe alsowriteA&Bas AB, andwe use D for material "implication." Parentheses areomittedunderthe conventions to of Church,Introduction Mathematical Logic:outermostparentheses are omitted;a dot mayreplacea left-hand parenthesis, the mateof whichis to be restoredat the end of the parenthetical partin whichthe dot occurs(otherwiseat the end of the formula);otherwiseparentheses areto be restored byassociation to the left. 2 Parry (20) presentsa notionof analytische alsoofferedas an explication Implikation, of "contained in the subject."Let ussaythatA is analytically relevantto B if everyvariable in B alsooccursin A. Then Parryunderstands occurring the matterin sucha waythat for A to "contain"B, A mustbe analytically relevantto B (andhe showshis systemhasthis property: in B arealsoin A). But thereis surely A-->Bis provableonlyif all the variables a sensein whichAvB is "contained" in A; viz., the sensein whichthe conceptSibling (whichis mostnaturally definedas Brother-or-Sister) is containedin the conceptBrother. Certainly"Allbrothersaresiblings"wouldhavebeenregarded as analyticby Kant(16). Anothersystemhavingthe sameproperty is thatof Hintikka(15). He writes:"Formulaewhicharetautologically equivalent by the propositional calculusareequivalent provided that they containoccurrences of exactlythe samefree variables,and so are expressions obtainedfromthemby replacingone or morefreeindividualvariablesby boundones." He then writesA )-B when this metalogical equivalence holds,and he lets A_>B abbreviateA< (A&B).Hintikkanowheresuggeststhat his -> is to be understoodas butwe takethe opportunity entailment, of pointingout thathis conditionis neithernecessarynorsufficientfortautological entailmenthood:B&A&AB satisfieshis condition, andA-,-AvBfails. ' We havenot defineda uniquenormalform,but it is readilyshownthatif onenormal formof A--B is a tautological entailment,thenall are,sincetheydifferonlyin the order of conjunctsanddisjuncts. ' We areindebtedto Neil Gallagher forpointingout an errorin an earlierattemptat interpreting the matrix. BIBLIOGRAPHY 1. Ackermann, Wilhelm."Begriindung einerstrengenImplikation," Journalof Symbolic Logic,21:113-28 (1956). 2. Anderson,AlanRoss.Completeness Theoremsfor the SystemsE of Entailmentand EQ of Entailmentwith Quantification. TechnicalReportNo. 6, Officeof Naval Research,GroupPsychology Branch,ContractSAR/Nonr-609(16),New Haven. (Reprintedin Zeitschriftfurmathematische Logik,6:201-16 (1959).) 3. Anderson, AlanRoss,andNuelD. Belnap,Jr."AModification of Ackermann's 'Rigorous Implication'"(abstract),Joumalof SymbolicLogic,23:457-58 (1958). in Ackermann's 4. . "Modalities 'RigorousImplication,'" journalof Symbolic Logic,24:107-11 (1959). 5. . "The PureCalculusof Entailment,"Journalof SymbolicLogic,forthcoming.

6. Anderson, AlanRoss,NuelD. Belnap,Jr.,andJohnR. Wallace."Independent Axiom Schematafor the Pure Theoryof Entailment,"Zeitschriftfur mathematische derMathematik, LogikundGrundlagen 6:93-95 (1960). 7. Belnap,Nuel D., Jr."PureRigorousImplicationas a sequenzen-kalkiil" (abstract), Journal of SymbolicLogic,24:282-83 (1959). 8. . A FormalAnalysisof Entailment.TechnicalReportNo. 7, Officeof NavalResearch,GroupPsychologyBranch,ContractNo. SAR/Nonr-609(16), NewHaven.

24 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

PHILOSOPHICAL STUDIES . "Entailment and Relevance," Journal of Symbolic Logic, 25:144-46 (1960). . "TautologicalEntailments" (abstract), Joumal of Symbolic Logic, 24: 316 (1959). Bennett, Jonathan F. "Meaning and Implication,"Mind, n.s., 63:451-63 (1954). Boehner,Philotheus.MedievalLogic. Chicago: Universityof Chicago Press, 1952. Fitch, FredericBrenton. SymbolicLogic. New York: Ronald Press, 1952. Geach, P. T. "Entailment,"AristotelianSociety SupplementaryVolume, 32:157-72 (1958). Hintikka,Jaakko."ExistentialPresuppositionsand ExistentialCommitments,"Journal of Philosophy,56(no. 3) :125-37 (1959). Kant, Immanuel. Critik der reinen Vernunft. Riga: Johann Friedrich Hartknoch, 1781. Kneale,William. "Truthsof Logic,"Proceedingsof the AristotelianSociety, 46:20734 (1945-46). Lewis, ClarenceIrving,and Cooper HaroldLangford.SymbolicLogic. New York and London: Appleton-Century,1932. Lewy, C. "Entailment," Aristotelian Society SupplementaryVolume, 32:123-42 (1958). Parry, William Tuthill. "Ein Axiomensystemfur eine neue Art von Implikation (analytischeImplikation)," Ergebnisseeines MathematischenKolloquiums,4:5-6 (1933). Popper,KarlR. "What Is Dialectic?"Mind, n.s., 49:403-26 (1940). . "Are ContradictionsEmbracing?"Mind, n.s., 52:47-50 (1943). . "New Foundationsfor Logic," Mind, n.s., 56:193-255 (1947). Prior,A. N. Formal Logic. New York: Oxford UniversityPress, 1955. Smiley, T. J. "Entailmentand Deducibility,"Proceedingsof the AristotelianSociety, 59:233-54 (1959). Strawson,P. F. "Review of von Wright [27]," Philosophical Quarterly,8:372-76 (1958). Von Wright, Georg Henrik. Logical Studies. London: Routledge and Kegan Paul, 1957. . "A Note on Entailment,"PhilosophicalQuarterly,9:363-65 (1959).

Infinite Analysis by WILLIAM TODD NORTHWESTERN UNIVERSITY

IN THiS articleI will examinethose analyseswhich assertthat a sentence of ordinarylanguageis equivalentto an infiniteconjunctionof othersentences. An exampleof this would be the usual phenomenalistanalysiswhere sentences about materialobjects are said to be equivalentto an infinite conjunctionof statementsabout sense experiences.Conversely,the behaviorist